The chain covering number of a poset with no infinite antichains

Abstract

The chain covering number $\Cov(P)$ of a poset $P$ is the least number of chains needed to cover $P$. For a cardinal $\nu$, we give a list of posets of cardinality and covering number $\nu$ such that for every poset $P$ with no infinite antichain, $\Cov(P)\geq ν$ if and only if $P$ embeds a member of the list. This list has two elements if $\nu$ is a successor cardinal, namely $[\nu {]}^2$ and its dual, and four elements if $\nu$ is a limit cardinal with $\cf(ν)$ weakly compact. For $\nu ={\aleph }_1$, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal $\nu$.

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